An evacuated glass vessel weighs `50.0 g` when empty, `148.0 g` when filled with a liquid of density `0.98 g mL^(-1)`, and `50.5 g` when filled with an ideal gas at `760 mm Hg` at `300 K`. Determine the molar mass of the gas.

Options

A. 100
B. 110
C. 123
D. 90

Explanation

To determine the molar mass of the gas in the evacuated glass vessel, we can follow these steps:

### Step 1: Calculate the mass of the ideal gas
The weight of the empty vessel is given as 50.0 g, and the weight of the vessel filled with the ideal gas is 50.5 g.

\[ \text{Mass of ideal gas} = \text{Weight of vessel with gas} - \text{Weight of empty vessel} \]
\[ \text{Mass of ideal gas} = 50.5 \, \text{g} - 50.0 \, \text{g} = 0.5 \, \text{g} \]

### Step 2: Calculate the mass of the liquid
The weight of the vessel filled with the liquid is given as 148.0 g.

\[ \text{Mass of liquid} = \text{Weight of vessel with liquid} - \text{Weight of empty vessel} \]
\[ \text{Mass of liquid} = 148.0 \, \text{g} - 50.0 \, \text{g} = 98.0 \, \text{g} \]

### Step 3: Calculate the volume of the liquid
The density of the liquid is given as 0.98 g/mL. We can use the formula for density to find the volume.

\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \implies \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
\[ \text{Volume} = \frac{98.0 \, \text{g}}{0.98 \, \text{g/mL}} = 100.0 \, \text{mL} = 0.1 \, \text{L} \]

### Step 4: Use the Ideal Gas Law to find the molar mass
The Ideal Gas Law is given by \( PV = nRT \), where:
- \( P \) = pressure in atm (760 mm Hg = 1 atm)
- \( V \) = volume in liters (0.1 L)
- \( n \) = number of moles
- \( R \) = universal gas constant (0.0821 L atm / (K mol))
- \( T \) = temperature in Kelvin (300 K)

First, we need to find the number of moles \( n \):
\[ n = \frac{\text{Mass}}{\text{Molar Mass}} \implies PV = \frac{\text{Mass}}{\text{Molar Mass}} RT \]

Rearranging gives us:
\[ \text{Molar Mass} = \frac{\text{Mass} \cdot R \cdot T}{P \cdot V} \]

Substituting the known values:
\[ \text{Molar Mass} = \frac{0.5 \, \text{g} \cdot 0.0821 \, \text{L atm/(K mol)} \cdot 300 \, \text{K}}{1 \, \text{atm} \cdot 0.1 \, \text{L}} \]

Calculating:
\[ \text{Molar Mass} = \frac{0.5 \cdot 0.0821 \cdot 300}{1 \cdot 0.1} \]
\[ \text{Molar Mass} = \frac{12.315}{0.1} = 123.15 \, \text{g/mol} \]

### Final Answer
The molar mass of the gas is **123.15 g/mol**.

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